PHYS-2402 Chapte 2 Lectue 2 Special Relativity 1. Basic Ideas Sep. 1, 2016 Galilean Tansfomation vs E&M y K O z z y K In 1873, Maxwell fomulated Equations of Electomagnetism. v Maxwell s equations descibe vey well all obseved E&M phenomena, but they ae not invaiant unde G.T.! E = ρ / ε0 B = 0 B Some odd things: E = t a chage in motion poduces a B-field, but a chage E at est does not. B = µ0 J + ε 0 µ0 the speed of light is the same in all IRFs, at odds t with Galilean velocity addition. O x x The Laws of Mechanics ae unchanged unde a Galilean tansfomation ε 0 µ0 = The Laws of E&M ae changed unde a Galilean tansfomation 6. Velocity Tansfomation 2. Consequences of Einstein s Postulates 7. Momentum & Enegy 1 c2 This bought into being an idea of a unique stationay RF (ethe), with espect to which all velocities ae to be measued What ae the options? At least one of the following statements must be wong: (a) the pinciple of elativity applies to 3 both mechanical and E&M phenomena; (b) Maxwell eqs ae coect; (c) G. T. ae coect. 3. The Loentz Tansfomation Equations 8. Geneal Relativity & a 1st Look at Cosmology 4. The Twin Paadox 9. The Light Baie 5. The Dopple Effects 10. The 4th Dimension Histoical Pespective Light is a wave. Waves equie a medium though which to popagate. Poposed solution: Light waves tavel in a medium called ethe (o aethe: fom Geek, meaning uppe ai). (Like sound waves in ai o wate waves in wate) The ethe becomes the absolute and unique fame of efeence whee Maxwell s equations hold. Maxwell s equations assume that light obeys the Newtonian-Galilean tansfomation. 4
Wave Popagation The Michelson-Moley Expeiment Expeiment designed to measue small changes in the speed of light was pefomed by Michelson (1852 1931, Nobel ) and Moley (1838 1923). Used an optical instument called an intefeomete that Michelson invented. Device was to detect the pesence of the ethe. Stat two light beams at the same point & time, then send them on two paths Paallel to Ethe: 5 one path paallel to the motion of the eath though the ethe one pependicula Measue the time diffeence it takes fo them to aive at the same point Pependicula to Ethe: 6
The Michelson-Moley Expeiment Paallel to Ethe: Pependicula to Ethe : Time fo each tip is given by Poblem: The speed of Eath moving though the ethe is ~10,000 times smalle than the speed of light. Instumentation of the peiod was not sensitive enough to measue such small timing diffeences. Solution: Obseve the intefeence between the two beams. 10 The Michelson-Moley Expeiment If the two beams ae moving at diffeent speeds, thee will be a fast am and a slow am. Deduce Δt fom obseved # finges. Outcomes of Michelson-Moley Expeiment Obsevation: No finge. T=peiod of light wave The speed of light was shown to be independent of the motion of the obseve. Helped cast away the ethe hypothesis E&M explained the oigin of light, but did not equie a medium fo its wave popagation 1907 Nobel Pize awaded to Michelson (1st Ameican Physicist) 11 Einstein s Postulates Big poblem at the tun of the centuy: 1. Michelson-Moley showed that the Galilean tansfomation did not hold fo Maxwell s equation. 2. Maxwell s equations could not be wong. 3. Galilean tansfomation did hold fo the laws of mechanics. 4. Einstein poposed a solution 12
Einstein s Pinciple of Relativity (1905) Einstein's Pinciple of Relativity (the fist postulate of the Special Theoy of Relativity): The laws of physics ae the same (covaiant) in all inetial efeence fames. The second postulate: The speed of light in vacuum is the same to all obseves, egadless of thei motion elative to the light souce. Thus, Maxwell s Equations ae in line with Einstein s Pinciple of Relativity. Conclusion: 2 nd Postulate: Light moves with the same speed (c) elative to all obseves Anna measues: Speed of light = c? Galilean Tansfomations must go. Einstein needed to find a new tansfomation (we will see this today) The idea of univesal and absolute time is wong! One has to come up with coect tansfomations that wok fo both mechanical and E&M phenomena (any speed up to ~c). Consequently, the laws of mechanics have to be modified to be covaiant unde new (coect) tansfomations. Obseves in all inetial systems measue the same value fo the speed of light in a vacuum. (c = 2.9979 x 10 8 m/s) Bob measues: Speed of light = c and not v+c The Ultimate Speed The speed of light has been defined to be exactly: c = 299 792 458 m/s Einstein s Postulates of Relativity: Light tavels at this ultimate speed, as do any massless paticles. No entity that caies enegy o infomation can exceed this speed limit. No paticle that does have a mass, can actually each c. Electons have been acceleated to at least 0.999 999 999 95 times the speed of light still less than c. 15 Light Souce, Medium and Michelson- Moley Expeiment (Discussion sessions, see Appendix A) Definition of an Event Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contaction
Consequence 1: Relative Simultaneity, o The absence of absolute simultaneity Simultaneous Flash Simultaneity is the popety of two events happening at the same time in a fame of efeence. An event is a physical occuence, independent of any efeence fame Accoding to Einstein's Theoy of Relativity, simultaneity is not an absolute popety between events; what is simultaneous in one fame of efeence will not necessaily be simultaneous in anothe. Simultaneous Aival => AN EVENT Simultaneous Aival!The same EVENT Fo Bob as well!
2 nd Postulate 2 nd Postulate The speed of light in vacuum is the same fo all inetial obseves, egadless of the motion of the souce. THEN: BUT THEN: Simultaneous Aival!AN EVENT Fo Bob as well! Simultaneous Emission is impossible fo Bob Simultaneous Aival!AN EVENT Fo Bob as well! Simultaneous Emission is impossible fo Bob Both emission and aival ae simultaneous Aival is simultaneous But Emission is not simultaneous Consequence 2: Time Dilation, o The absence of absolute time Two events occuing at the same location in one fame will be sepaated by a longe time inteval in a fame moving elative to the fist. This is not an optical illusion. Space and time ae diffeent fo all obseves in elative motion.
The time fo the light to etun:!= 2"/c The time fo the light to etun:! = 2"/c Longe Path (L) + 2 nd Postulate PRECISELY => T = 2 L/c L 2 = H 2 + (v T/2) 2 # = 2"/c The time fo the light to etun:! > 2"/c L The time fo the light to etun: # > 2"/c Longe Path + 2 nd Postulate T = # (1 (v/c) 2 ) -1/2
Consequence 3: Length Contaction Consequence 3: Length Contaction Evidence fo Relativistic Effects The length of an object in a fame though which the object moves is smalle than its length in the fame which it is at est. o Length of a moving object contacts in the diection of motion. Consequence 3: Loentz Length Contaction, o The absence of absolute distance Muons: subatomic paticles poduced by cosmic-ays in the uppe atmosphee. L = L0 (1 (v/c)2 )1/2 29 30 Consequence 3: Length Contaction Evidence fo Relativistic Effects: Cosmic Ray Muons Anna and Bob agee on events but disagee on the space and time pattens of those events. Ceated in the uppe atmosphee (3 km above gound) when potons impact the atmosphee. A muon s lifetime has been measued to be 2.2 μs. They tavel at nealy the speed of light. How fa do we expect that can they tavel duing thei life-time? We need a way to tansfom what they see that fits within the postulates of special elativity. Einstein needed to find a new tansfomation (the old one being the Galilean tansfomation). It must fit both the laws of mechanics & Maxwell s E&M equations. It must allow time to be elative. D = vt = (0.98)(3 x 108 m/s) x (2.2 x 10-6 s) = 647 m This calculation tells us that muons should not make it to Eath. But we know that they do. (moe late) Muon will each the suface; D = 3.2 km (> 3km) 31 32
Conside the tansfomation between fames S and S of an object with no net foces moving at a constant velocity. Geneal elationship elating both fames. Note: We equie only that the tansfomation be linea. It is not equied that time be absolute. In each fame (S & S ), the space, speed and time fo the object ae elated. Fom Need to detemine A, B, C, & D 33 Detemine the coefficients by consideing 3 special cases. 34 CASE 1: The object is pinned to the oigin of S. What is the speed and position as seen by an obseve in the S fame? CASE 2: The object is pinned to the oigin of S. What is the speed and position as seen by an obseve in the S fame? What is the speed and position as seen by an obseve in the S fame? What is the speed and position as seen by an obseve in the S fame? Going back to ou linea elationships, we have Fom ou linea elationships, Since, Sub t = Dt and x = 0 We have...(1) 35 36
CASE 3: Assume the object is moving at the speed of light. One moe step: Rewite ou linea tansfomation. Fom ou linea elationships, Thus, Recall A = D Since, In ode to find A, we solve these equations fo x and t. 37 38 Solve fo x fist. Solve fo t. Substitute (2) into (1) 39 40
Examine ou equations: We can use γ to wite ou tansfomations. Since the two fames ae in constant elative motion w..t each othe, the ONLY diffeence between the fames is that S moves at +v elative to fame S, while S moves at v elative to S. So, above eqs. have to be identical except fo the sign of v. Fame S Fame S Loentz Facto 41 42